Anomalous electrical and frictionless flow conductance in complex networks

We study transport properties such as electrical and frictionless flow conductance on scale-free and Erdo{double acute}s-Rényi networks. We consider the conductance G between two arbitrarily chosen nodes where each link has the same unit resistance. Our theoretical analysis for scale-free networks predicts a broad range of values of G, with a power-law tail distribution Φ_SF (G) ∼ G^- g_G, where g_G = 2 λ - 1, where λ is the decay exponent for the scale-free network degree distribution. We confirm our predictions by simulations of scale-free networks solving the Kirchhoff equations for the conductance between a pair of nodes. The power-law tail in Φ_SF (G) leads to large values of G, thereby significantly improving the transport in scale-free networks, compared to Erdo{double acute}s-Rényi networks where the tail of the conductivity distribution decays exponentially. Based on a simple physical 'transport backbone' picture we suggest that the conductances of scale-free and Erdo{double acute}s-Rényi networks can be approximated by c k_A k_B / (k_A + k_B) for any pair of nodes A and B with degrees k_A and k_B. Thus, a single quantity c, which depends on the average degree over(k, -) of the network, characterizes transport on both scale-free and Erdo{double acute}s-Rényi networks. We determine that c tends to 1 for increasing over(k, -), and it is larger for scale-free networks. We compare the electrical results with a model for frictionless transport, where conductance is defined as the number of link-independent paths between A and B, and find that a similar picture holds. The effects of distance on the value of conductance are considered for both models, and some differences emerge. Finally, we use a recent data set for the AS (autonomous system) level of the Internet and confirm that our results are valid in this real-world example.